Week 9 - Sources of Magnetic Fields October 22, 2012 Ampere was the Newton of Electricity. James C. Maxwell Exercise 9.1: Discussion Quesitons a) The text in the book discusses the case of an infinite long conducting wire carrying a current. Of course, nothing is really infinite. How do you decide whether a wire is long enough to be considered infinite? A wire can be considered infinite to a very good approximation if it s length is much greater than any other relevant dimension in the problem. For example, suppose we are interested in the field at point P a distance a from a wire of length l where a/l 1, then since a is so small compared to l, l can for the purpose of finding the magnetic field at P be considered infinite. If you stand at P the wire it will look really long to you. However, if a/l 1 the wire will look really short and the infinite approximation is no good. b) Pairs of conductors carrying current into or out of the power supply components of electronic equipment are sometimes twisted together to reduce effects of magnetic fields. Why does this help? The problem is when one of the cables is affected more by electromagnetic radiation than the other. Most electronic devices use the voltage difference across the two cables to transmit signals. 1
If both are affected by the same amount from an external electromagnetic field, there is no problem because the difference will still be the same. If however one is affected more than the other, noise is introduced. This becomes a problem when the noise source is close to the wires. Then the electromagnetic field may differ much between the ingoing and outgoing wire, and noise is introduced. This is a big problem when multiple pairs of telecommunication cables are in close proximity, leading to noise between the different pairs. This effect is often increased because telecommunication cables may lie close to each other for several kilometers. By twisting the wires, each wire is exposed to as much of the noise from the other wires as the other. This is for instance used in network cables. You can read more about the topic on Wikipedia. c) What features of atomic structure determine whether an element is diamagnetic or paramagnetic? Explain. Whether it has unpaired electrons or not. d) A cylinder of iron is placed so that it is free to rotate around its axis. Initially the cylinder is at rest in a magnetic field applied to the cylinder so that it is magnetized in a direction parallel to its axis. If the direction of the external field is suddenly reversed, the direction of magnetization will also reverse and the cylinder will begin to rotate. (This is called the Einstein-de Haas effect.). Explain why the cylinder starts to rotate. We have seen that the atoms in the material have a magnetic moment µ due to the orbiting electrons around the nucleus. This magnetic moment points in the direction perpendicular to the plane of the orbit determined by the right hand rule. However an electron does also have mass and a particle with mass in orbit does also have an associated angular momentum which points in the exact same direction as the magnetic moment. Therefore this two quantities are proportional to each other and we have earlier found that µ = e 2m L (1) where L is the orbital angular momentum of the orbiting mass. So as the cylinder get magnetized and acquires a magnetic moment µ = V M it also acquires an angular momentum L 0. It is due to this proportionality that a magnetized cylinder also acquires an angular momentum such that it begins to rotate. Exercise 9.2: Magnetic Field from a Lightning Bolt Lightning Bolts carry on average currents at about 30 100 ka. We can model such a strike as a very long straight wire. a) If you were unfortunate enough to be 5.0 m away from such a lightning bolt, how large a magnetic would you experience? B = 4π 10 7 100 10 3 2π 5 T 10 6 10 5 10 T = 10 2 T. (2) 2
b) How does this field compare to one you would experience by being 5.0 cm away from a long wire carrying an average household current of 10 A? B = 10 6 T (3) Exercise 9.3: Strongest Magnetic Field Ever Created Scientists at Los Alamos National Laboratory in America has succeeded in creating the strongest magnetic field ever produced in a laboratory. They reached a stunning 100.75 T using an electromagnet. Assuming they used a solenoid electromagnet with about 10 4 turns and of a length on the order of 1 m. What current did they need in order to produce this magnetic field? I = B µ 0 n = 10 2 4π 10 7 10 4 A 102 10 10 7 10 4 A = 10 4 A. (4) Exercise 9.4: A Magnetic Field Engineer An engineer specializing in the creation of neophyte magnets claims he has a design which can produce a magnetic field in vacuum which points everywhere in the x-direction and which increases proportional to x. In other words he claims that his magnetic field has the form B = B 0 (x/a)î. Show that his claim is impossible. Hint: Use Gauss law for magnetism on differential form B = B 0 x a î = B 0 a 0 (5) Exercise 9.5: Coaxial Cable A solid conductor with radius a is supported by insulating disks on the axis of a conducting tube with inner radius b and outer radius c. A section of this long cable is shown in figure 1. The central conductor and tube carry equal currents I in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression of the magnitude of the magnetic field for all radii r away from the axis of the cable. Can you now name one advantage coaxial cables have over other types of cables? 3
Figure 1 B = µ 0I r 0 < r a. (6) 2πa2 B = µ 0I 2πr a < r b. (7). ) 2 B = µ 0I 1 ( r c 2πr 1 ( ) b 2 b < r c (8) c There is no magnetic field outside the cable! B = 0 r > c (9) Figure 2 Exercise 9.6: Field from a Line Segment Figure 2 shows a line segment of length 2L carrying a current I. Use Biot-Savart s law to find the field at the point P. Show that you obtain the familiar expression B = µ 0 I/2πr when the length of the rod goes to infinity. Why can t you use Amperes law in this situation of a finite rod? B = µ 0Iˆk 4πy (cos θ 0 cos θ 1 ) (10) 4
For an infinite rod, B = µ 0I 2πy ˆk. (11) 5